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stat Midterm front

stat flashcards

– Descriptive: primary purpose is to describe some aspect of the data
Inferential: primary purpose is to infer (to estimate or to make a decision, test a hypothesis)

All inferential statistics have the following in common:

– use of some descriptive statistic
– use of probability
– potential for estimation
– sampling variability
– sampling distributions
– use of a theoretical distribution
– two hypotheses, two decisions, two types of error

• no manipulation
• minimal control of EV
• predictive relationship between PV and CV

Stem and Leaf Display

• The first digit(s) of a score form the stem, the last digit(s) form the leaf.
• We want 10-20 total number of stems.
• Number of stems per digit depends on total number of stems: can do 1, 2, or 5 stems per digit.

Description With Statistics
Aspects or characteristics of data that we can describe are:

– Middle
– Spread
– Skewness
– Kurtosis

Other words that describe Middle

central tendency, location, center

Statistics that Measure middle are:

mean, median, mode
• “Middle” is the aspect of data
we want to describe.
• We describe/measure the middle of data in a population with the parameter m (‘mu’); we usually don’t know m, so we estimate it with X-bar.

Other words that describe Spread

variability, dispersion, skatter

Statistics that Measure spread are:

range, variance, standard deviation, midrange
• “Spread” is the aspect of data we want to describe.
• Any statistic that describes/measures spread should have these characteristics: it should
– Equal zero when the spread is zero.
– Inc

-The sample mean is the sum of the scores divided by the number of scores, and is symbolized by X-bar, X = SX/N
-For example, for X1=4, X2=1, X3=7, N=3, SX=12 and X = SX/N = 12/3 = 4
• Characteristics:
– X-bar is the balance point

Sample Median

• The median is the middle of the ordered scores, and is symbolized as X50.
• Median position (as distinct from the median itself) is (N+1)/2 and is used to find the median.
• Example: X1=4, X2=1, X3=7, then N=3.
• Characteristic

Sample Mode

• The mode is the most frequent score.
• Examples:
– 1 1 4 7, the mode is 1.
– 1 1 4 7 7, there are two modes, 1 and 7.
– 1 4 7, there is no mode.
• Characteristics:
– Has problems: more than one, or none; maybe not in the mid

• Formula is MR=UH-LH
– UH=upper hinge
– LH=lower hinge
– Hinges cut off 25% of the data in each tail
• Hinge position is ([median position]+1)/2.
– [median position] is the whole number part of the median position (remember, median p

Hinge position

([median position]+1)/2
– [median position] is the whole number part of the median position (remember, median pos.=(N+1)/2)
• Use hinge position to count in from the tails to find the hinges.

• The aspect of the data we want to describe/measure is relative position. • z scores are statistics that describe the relative position of something in its distribution.

Z score formula

z is something minus its mean divided by its standard deviation.

z score characteristics

– The mean of a distribution of z scores is zero.
– The variance of a distribution of z scores is one.
– The shape of a distribution of z scores is reflective, the shape is the same as the shape of the distribution of the Xs.

Characteristics of Normal Distributions

– Symmetric, continuous, unimodal.
– Bell-shaped.
– Scores range from -¥ to +¥ .
– Mean, median, and mode are all the same value.
– Each distribution has two parameters, m and s².

Use of Z score

• We use this distribution to get probabilities associated with a z score (probability, proportion, and area under the curve are synonymous).
- look up z in table to find probabilities.

Correlation

– Defined as the degree of linear relationship between X and Y. – Is measured/described by the statistic r.

Regression

– Is concerned with the prediction of Y from X Forms a prediction equation to predict Y from X
Uses the formula for a straight line, Y’=bX+a.
– Y’ is the predicted Y score on the criterion variable.
– b is the slope, b=DY/ D X=rise/run.
–

r=

r=SzXzY/N, the average product of z scores for X and Y
– Works with two variables, X and Y
– -1<r<1, r measures positive or negative relationships
– Measures only the degree of linear relationship
– r2=proportion of variability in Y that is e

r2=

proportion of variability in Y that is explained by X.

Correlation: Undefined

If there is no spread in X or Y, then r is undefined. Note that any z is undefined if the standard deviation is zero, and r=SzXzY/N.

Population correlation coefficient,

r (rho)

regression cont.

• Linear only.
• Generalize only for X values in
your sample.
• Actual observed Y is different from Y’ by an amount called error, e, that is, Y=Y’+e.
• Error in regression is e=Y-Y’.
• Many different potential regression

Line of Best Fit

The statistics b and a are computed so as to minimize the sum of squared errors, – Se2=S(Y-Y’)2 is a minimum. – This is called the Least Squares Criterion.

• p(A|B)=(number in [A and B])/(number in B)
• The probability of A in the redefined (reduced) sample space of B.

Big 3 Probability Rules

1. independence 2. mulitplication, mutually exclusive 3.) addition

Independence (1)

events A and B are independent if
• p(A|B)=p(A)
• The A probability is not changed by
reducing the sample space to B.

Multiplication (And) Rule (2)

• p(A and B)=p(A)p(B|A)=p(A|B)p(B)

Mutually exclusive:

• Events A and B do not have any elementary events in common.
• Events A and B cannot occur simultaneously.
• p(A and B)=0

Addition (Or) Rule (3)

p(A or B)=p(A)+p(B)-p(A and B)

The sampling distribution of X-bar

– Has the purpose of any sampling distribution: to obtain probabilities…
– Has the definition of any sampling distribution: the distribution of a statistic.
– Has specific characteristics:
• Mean: mX = m
• Variance: s2X =s2/N
• Shape i

Hypothesis testing

is the process of testing tentative guesses about relationships between variables in populations. These relationships between variables are evidenced in a statement , a hypothesis, about a population parameter.

Test statistic

a statistic used only for the purpose of testing hypotheses; e.g. zX

Assumptions

conditions placed on a test statistic necessary for its valid use in hypothesis testing;– for zX, the assumptions are that the population is normal in shape and that the observations are independent.

Null hypothesis

the hypothesis that we test; Ho.

Alternative hypothesis

where we put what we believe; H

Significance level

he standard for what we mean by a “small” probability in hypothesis testing; a.
The significance level is the small probability used in hypothesis testing to determine an unusual event that leads you to reject Ho.
– The significance level is sym

Direcetional v. Non-Directional Hypothesis

>,<, or =
• Directional hypotheses specify a particular direction for values of the parameter.
– IQ of deaf children example: Ho: m>100, H1: m<100.
• Non-directional hypotheses do not specify a particular direction for values of the paramet

One- and two-tailed tests

– A one-tailed test is a statistical test that uses only one tail of the sampling distribution of the test statistic.
– A two-tailed test is a statistical test that uses two tails of the sampling distribution of the test statistic.

Critical values

values of the test statistic that cut off a or a/2 in the tail(s) of the theoretical reference distribution.

Rejection values

the values of the test statistic that lead to rejection of Ho

p-Value Decision Rules

• Reject Ho if
– ½ the SAS p-value <a, and
– the observed zX is in the tail specified by H1.